Cusp universality for correlated random matrices

Erdös L, Henheik SJ, Riabov V. 2025. Cusp universality for correlated random matrices. Communications in Mathematical Physics. 406(10), 253.

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Abstract
For correlated real symmetric or complex Hermitian random matrices, we prove that the local eigenvalue statistics at any cusp singularity are universal. Since the density of states typically exhibits only square root edge or cubic root cusp singularities, our result completes the proof of the Wigner–Dyson–Mehta universality conjecture in all spectral regimes for a very general class of random matrices. Previously only the bulk and the edge universality were established in this generality (Alt et al. in Ann Probab 48(2):963–1001, 2020), while cusp universality was proven only for Wigner-type matrices with independent entries (Cipolloni et al. in Pure Appl Anal 1:615–707, 2019; Erdős et al. in Commun. Math. Phys. 378:1203–1278, 2018). As our main technical input, we prove an optimal local law at the cusp using the <jats:italic>Zigzag strategy</jats:italic>, a recursive tandem of the characteristic flow method and a Green function comparison argument. Moreover, our proof of the optimal local law holds uniformly in the spectrum, thus we also provide a significantly simplified alternative proof of the local eigenvalue universality in the previously studied bulk (Erdős et al. in Forum Math. Sigma 7:E8, 2019) and edge (Alt et al. in Ann Probab 48(2):963–1001, 2020) regimes.
Publishing Year
Date Published
2025-09-01
Journal Title
Communications in Mathematical Physics
Publisher
Springer Nature
Acknowledgement
We thank Giorgio Cipolloni for many productive discussions and the anonymous referees for several useful suggestions and spotting some typos. Open access funding provided by Institute of Science and Technology (IST Austria).
Volume
406
Issue
10
Article Number
253
ISSN
eISSN
IST-REx-ID

Cite this

Erdös L, Henheik SJ, Riabov V. Cusp universality for correlated random matrices. Communications in Mathematical Physics. 2025;406(10). doi:10.1007/s00220-025-05417-z
Erdös, L., Henheik, S. J., & Riabov, V. (2025). Cusp universality for correlated random matrices. Communications in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s00220-025-05417-z
Erdös, László, Sven Joscha Henheik, and Volodymyr Riabov. “Cusp Universality for Correlated Random Matrices.” Communications in Mathematical Physics. Springer Nature, 2025. https://doi.org/10.1007/s00220-025-05417-z.
L. Erdös, S. J. Henheik, and V. Riabov, “Cusp universality for correlated random matrices,” Communications in Mathematical Physics, vol. 406, no. 10. Springer Nature, 2025.
Erdös L, Henheik SJ, Riabov V. 2025. Cusp universality for correlated random matrices. Communications in Mathematical Physics. 406(10), 253.
Erdös, László, et al. “Cusp Universality for Correlated Random Matrices.” Communications in Mathematical Physics, vol. 406, no. 10, 253, Springer Nature, 2025, doi:10.1007/s00220-025-05417-z.
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